# AP Physics C: Mechanics Practice (Newton s Laws including friction, resistive forces, and centripetal force).

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1 AP Physics C: Mechanics Practice (Newton s Laws including friction, resistive forces, and centripetal force). 1981M1. A block of mass m, acted on by a force of magnitude F directed horizontally to the right as shown above, slides up an inclined plane that makes an angle θ with the horizontal. The coefficient of sliding friction between the block and the plane is µ. a. On the diagram of the block below, draw and label all the forces that act on the block as it slides up the plane. b. Develop an expression in terms of m, θ, F, µ, and g, for the block s acceleration up the plane. c. Develop an expression for the magnitude of the force F that will allow the block to slide up the plane with constant velocity. What relation must θ and µ satisfy in order for this solution to be physically meaningful?

2 1984M1. An amusement park ride consists of a rotating vertical cylinder with rough canvas walls. The floor is initially about halfway up the cylinder wall as shown above. After the rider has entered and the cylinder is rotating sufficiently fast, the floor is dropped down, yet the rider does not slide down. The rider has mass of 50 kilograms, the radius R of the cylinder is 5 meters, the angular velocity of the cylinder when rotating is 2 radians per second, and the coefficient of static friction between the rider and the wall of the cylinder is 0.6. a. On the diagram below. draw and identify the forces on the rider when the system is rotating and the floor has dropped down. b. Calculate the centripetal force on the rider when the cylinder is rotating and state what provides that force. c. Calculate the upward force that keeps the rider from falling when the floor is dropped down and state what provides that force. d. At the same rotational speed, would a rider of twice the mass slide down the wall? Explain your answer.

3 1984M3. A small body of mass m located near the Earth s surface falls from rest in the Earth's gravitational field. Acting on the body is a resistive force of magnitude kmv, where k is a constant and v is the speed of the body. a. On the diagram below, draw and identify all of the forces acting on the body as it falls. b. Write the differential equation that represents Newton's second law for this situation. c. Determine the terminal speed v T of the body. d. Integrate the differential equation once to obtain an expression for the speed v as a function of time t. Use the condition that v = 0 when t= 0. e. On the axes provided above, draw a graph of the speed v as a function of time t.

4 1986M1. The figure above shows an 80-kilogram person standing on a 20-kilogram platform suspended by a rope passing over a stationary pulley that is free to rotate. The other end of the rope is held by the person. The masses of the rope and pulley are negligible. You may use g = 10 m/ s 2. Assume that friction is negligible, and the parts of the rope shown remain vertical. a. If the platform and the person are at rest, what is the tension in the rope? The person now pulls on the rope so that the acceleration of the person and the platform is 2 m/s 2 upward. b. What is the tension in the rope under these new conditions? c. Under these conditions, what is the force exerted by the platform on the person? After a short time, the person and the platform reach and sustain an upward velocity of 0.4 m/ s. d. Determine the power output of the person required to sustain this velocity.

5 1988M1. A highway curve that has a radius of curvature of 100 meters is banked at an angle of 15 as shown above. a. Determine the vehicle speed for which this curve is appropriate if there is no friction between the road and the tires of the vehicle. On a dry day when friction is present, an automobile successfully negotiates the curve at a speed of 25 m/s. b. On the diagram above, in which the block represents the automobile, draw and label all of the forces on the automobile. c. Determine the minimum value of the coefficient of friction necessary to keep this automobile from sliding as it goes around the curve.

6 1990M1. An object of mass m moving along the x-axis with velocity v is slowed by a force F = -kv, where k is a constant. At time t = 0, the object has velocity v o at position x = 0, as shown above. a. What is the initial acceleration (magnitude and direction) produced by the resistance force? b. Derive an equation for the object's velocity as a function of time t, and sketch this function on the axes below. Let a velocity directed to the right be considered positive. c. Derive an equation for the distance the object travels as a function of time t and sketch this function on the axes below. d. Determine the distance the object travels from t = 0 to t =.

7 1993M2. A car of mass m, initially at rest at time t = 0, is driven to the right, as shown above, along a straight, horizontal road with the engine causing a constant force F o to be applied. While moving, the car encounters a resistance force equal to -kv, where v is the velocity of the car and k is a positive constant. a. The dot below represents the center of mass of the car. On this figure, draw and label vectors to represent all the forces acting on the car as it moves with a velocity v to the right. b. Determine the horizontal acceleration of the car in terms of k, v, F o, and m. c. Derive the equation expressing the velocity of the car as a function of time t in terms of k, F o, and m.

8 1996M2. A 300- kg box rests on a platform attached to a forklift, shown above. Starting from rest at at time = 0, the box is lowered with a downward acceleration of 1.5m/s 2 a. Determine the upward force exerted by the horizontal platform on the box as it is lowered. At time t = 0, the forklift also begins to move forward with an acceleration of 2 m/s 2 while lowering the box as described above. The box does not slip or tip over. b. Determine the frictional force on the box. c. Given that the box does not slip, determine the minimum possible coefficient of friction between the box and the platform. d. Determine an equation for the path of the box that expresses y as a function of x (and not of t), assuming that, at time t = 0, the box has a horizontal position x = 0 and a vertical position y = 2 m above the ground, with zero velocity. e. On the axes below sketch the path taken by the box

9 1998M3. Block 1 of mass m 1 is placed on block 2 of mass m 2 which is then placed on a table. A string connecting block 2 to a hanging mass M passes over a pulley attached to one end of the table, as shown above. The mass and friction of the pulley are negligible. The coefficients of friction between blocks 1 and 2 and between block 2 and the tabletop are nonzero and are given in the following table. Express your answers in terms of the masses, coefficients of friction, and g, the acceleration due to gravity. a. Suppose that the value of M is small enough that the blocks remain at rest when released. For each of the following forces, determine the magnitude of the force and draw a vector on the block provided to indicate the direction of the force if it is nonzero. i. The normal force N 1 exerted on block 1 by block 2 m 1 ii. The friction force f 1 exerted on block 1 by block 2 m 1 iii. The force T exerted on block 2 by the string m 2 iv. The normal force N 2 exerted on block 2 by the tabletop m 2 v. The friction force f 2 exerted on block 2 by the tabletop m 2 b. Determine the largest value of M for which the blocks can remain at rest. c. Now suppose that M is large enough that the hanging block descends when the blocks are released. Assume that blocks 1 and 2 are moving as a unit (no slippage). Determine the magnitude a of their acceleration. d. Now suppose that M is large enough that as the hanging block descends, block 1 is slipping on block 2. Determine each of the following. i. The magnitude a 1 of the acceleration of block 1

10 ii. The magnitude a 2 of the acceleration of block M2. A rubber ball of mass m is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused by the air). The drag force on the ball has magnitude bv 2, where b is a constant drag coefficient and v is the instantaneous speed of the ball. The drag coefficient b is directly proportional to the cross- sectional area of the ball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approaches a constant value called the terminal speed. a. On the figure below, draw and label all the forces on the ball at some instant before it reaches terminal speed. b. State whether the magnitude of the acceleration of the ball of mass m increases, decreases, or remains the same as the ball approaches terminal speed. Explain. c. Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t, the given quantities, and fundamental constants. d. Determine the terminal speed vt in terms of the given quantities and fundamental constants. e. Determine the energy dissipated by the drag force during the fall if the ball is released at height h and reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental constants.

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